Subgroups — #10610390

Order 640.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66162	C₂	1	1	yes	maximal, characteristic
2	C₂×#4094	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#65408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3952	C₂	1	1	yes	maximal, characteristic, complemented
7	#65390	C₂	1	1	yes	maximal, characteristic, complemented
8	#65019	C₂	1	2	yes	maximal, complemented
9	#65018	C₂	1	2	yes	maximal, complemented
10	#65017	C₂	1	2	yes	maximal, complemented
11	#65014	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
16	#4094	—	2	4	no	—
17	#4092	C₄	1	1	yes	characteristic
18	#4092	C₄	1	1	yes	characteristic
19	#4084	C₄	1	1	yes	characteristic
20	#4084	C₄	1	1	yes	characteristic
21	#4083	C₄	1	2	yes	—
22	#4081	C₄	1	2	yes	—
23	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
25	#4072	C₂²	1	1	yes	characteristic
26	C₂×#681	C₂²	1	1	yes	characteristic
27	C₂×#681	C₂²	1	2	yes	complemented
28	C₂²×#168	C₂²	1	1	yes	characteristic
29	C₂²×#168	C₂²	1	2	yes	complemented
30	D₅×M₁₆	—	2	4	no	—
31	C₂×#676	C₂²	1	2	yes	—
32	C₂×C₈×D₅	C₂²	1	2	yes	—
33	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	#3952	C₂²	1	1	yes	characteristic
36	#3952	C₂²	1	1	yes	characteristic
37	#3952	C₂²	1	1	yes	characteristic
38	#3952	C₂²	1	1	yes	characteristic
39	#3947	C₂²	1	2	yes	complemented
40	#3946	C₂²	1	2	yes	complemented
41	#3946	C₂²	1	2	yes	complemented
42	C₈×Dic₅	C₂²	1	2	yes	complemented
43	#2933	—	5	1	no	maximal, Sylow
44	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
45	C₅×#41	—	2	2	no	—
46	C₅×#41	—	2	2	no	—
47	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂³	1	2	yes	—
51	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
54	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	—	2	2	no	—
56	C₂×C₄×D₅	—	2	2	no	—
57	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
61	C₅×M₁₆	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂³	1	2	yes	—
66	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
69	#685	C₂×C₄	1	2	yes	—
70	#685	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
73	#683	C₂×C₄	1	2	yes	—
74	#683	C₂×C₄	1	2	yes	—
75	C₄×Dic₅	C₂³	1	2	yes	—
76	#681	—	2	4	no	—
77	#681	D₄	1	4	yes	complemented
78	C₂×#168	—	2	2	no	—
79	C₂×#168	—	2	4	no	—
80	C₂×#168	C₂³	1	1	yes	characteristic
81	C₂×#168	C₂³	1	1	yes	characteristic
82	C₂×#168	C₂³	1	2	yes	—
83	#676	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	C₂×#131	—	5	1	no	—
86	C₂²×M₁₆	—	5	1	no	—
87	#525	—	5	1	no	—
88	#436	—	5	2	no	—
89	#435	—	5	2	no	—
90	#435	—	5	2	no	—
91	C₈×D₄	—	5	2	no	—
92	#424	—	5	1	no	—
93	#408	D₅	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊C₈	—	5	1	no	—
95	#405	—	5	1	no	—
96	C₂³×C₅	C₂²×C₄	1	2	yes	—
97	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₅	—	2	2	no	—
99	C₂²×D₅	—	2	2	no	—
100	C₂²×D₅	—	4	1	no	—
101	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
103	C₅×Q₈	—	2	2	no	—
104	C₅×D₄	—	2	4	no	—
105	C₅×D₄	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₅	#41	1	2	yes	—
112	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₅	—	2	4	no	—
114	C₂²⋊D₅	—	4	2	no	—
115	C₂×Dic₅	—	2	2	no	—
116	C₂×Dic₅	C₂²×C₄	1	2	yes	—
117	C₂×Dic₅	C₂²×C₄	1	2	yes	—
118	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
120	C₄×D₅	—	2	2	no	—
121	C₄×D₅	—	2	2	no	—
122	C₄×D₅	—	2	1	no	—
123	C₄×D₅	—	2	1	no	—
124	C₄×D₅	—	2	1	no	—
125	C₄×D₅	—	2	1	no	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	2	1	no	—
129	#168	—	2	1	no	—
130	#168	—	2	1	no	—
131	#168	C₂×D₄	1	4	yes	—
132	C₂×#41	C₂×D₅	1	1	yes	characteristic
133	C₂³×C₄	—	5	1	no	—
134	#131	—	10	4	no	—
135	C₂×M₁₆	—	5	1	no	—
136	C₂×M₁₆	—	5	2	no	—
137	C₂×M₁₆	—	5	2	no	—
138	C₂×M₁₆	—	10	4	no	—
139	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
140	C₂²×C₈	—	5	1	no	—
141	C₂²×C₈	—	5	2	no	—
142	C₂²×C₈	—	5	2	no	—
143	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
144	#123	—	5	2	no	—
145	C₂²⋊Q₈	—	5	1	no	—
146	C₂²⋊Q₈	—	5	1	no	—
147	#121	—	5	1	no	—
148	#121	—	5	1	no	—
149	#120	—	5	2	no	—
150	C₄×D₄	—	5	2	no	—
151	C₄×D₄	—	5	2	no	—
152	#117	—	5	1	no	—
153	C₄⋊C₈	—	5	2	no	—
154	C₄⋊C₈	—	5	2	no	—
155	C₂²⋊C₈	—	5	1	no	—
156	C₂²⋊C₈	—	5	1	no	—
157	C₂²⋊C₈	—	5	1	no	—
158	C₂²⋊C₈	—	5	1	no	—
159	C₂²⋊C₈	C₂×D₅	1	2	yes	complemented
160	C₂²⋊C₈	C₂×D₅	1	2	yes	complemented
161	#97	—	5	2	no	—
162	C₄×C₈	—	5	2	no	—
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	C₄×D₄	1	2	yes	—
169	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₂×D₅	—	2	1	no	—
171	C₂×D₅	—	2	1	no	—
172	C₂×D₅	—	2	1	no	—
173	C₂×D₅	—	2	1	no	—
174	C₂×D₅	—	2	1	no	—
175	C₂×D₅	—	4	4	no	—
176	C₂×D₅	#131	1	2	yes	complemented
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	C₄×D₄	1	2	yes	—
179	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₅	C₂×#41	1	1	yes	characteristic
181	Dic₅	—	2	1	no	—
182	Dic₅	—	2	1	no	—
183	Dic₅	—	2	2	no	—
184	Dic₅	—	2	2	no	—
185	C₂⁴	—	5	1	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
189	C₂×D₄	—	5	2	no	—
190	C₂×D₄	—	5	2	no	—
191	C₂×D₄	C₄×D₅	1	2	yes	—
192	C₂×D₄	C₄×D₅	1	1	yes	characteristic
193	C₂²×C₄	—	5	1	no	—
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	10	2	no	—
199	C₂²×C₄	—	10	2	no	—
200	C₂²×C₄	C₂²×D₅	1	2	yes	—
201	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	5	4	no	—
204	M₁₆	—	10	4	no	—
205	M₁₆	—	10	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	—	10	4	no	—
211	C₂×C₈	—	10	4	no	—
212	C₂×C₈	—	10	2	no	—
213	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
214	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₅	1	2	yes	—
216	C₄⋊C₄	—	5	1	no	—
217	C₄⋊C₄	—	5	1	no	—
218	C₄⋊C₄	—	5	2	no	—
219	C₄⋊C₄	—	5	2	no	—
220	C₂²⋊C₄	—	5	2	no	—
221	C₂²⋊C₄	—	5	2	no	—
222	C₂²⋊C₄	—	5	2	no	—
223	C₂²⋊C₄	—	5	2	no	—
224	C₂²⋊C₄	—	5	2	no	—
225	C₄²	—	5	2	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	4	2	no	—
228	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
229	C₂×C₅	C₂×#131	1	1	yes	characteristic
230	C₂×C₅	#568	1	1	yes	characteristic
231	D₅	—	2	2	no	—
232	D₅	—	4	1	no	—
233	C₂³	—	5	1	no	—
234	C₂³	—	5	1	no	—
235	C₂³	—	10	2	no	—
236	C₂³	—	10	2	no	—
237	C₂³	—	20	1	no	—
238	C₂³	C₂×C₄×D₅	1	2	yes	—
239	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	2	no	—
243	D₄	—	10	4	no	—
244	D₄	—	20	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	5	2	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	10	2	no	—
252	C₂×C₄	—	10	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	10	1	no	—
255	C₂×C₄	—	10	1	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
260	C₂×C₄	D₄×D₅	1	2	yes	—
261	C₂×C₄	#711	1	2	yes	—
262	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	5	4	no	—
267	C₈	—	10	1	no	—
268	C₈	—	10	1	no	—
269	C₅	#2933	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	5	2	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	—	10	1	no	—
279	C₂²	—	10	1	no	—
280	C₂²	—	10	1	no	—
281	C₂²	—	20	4	no	—
282	C₂²	#4027	1	2	yes	—
283	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	10	1	no	—
286	C₄	—	10	1	no	—
287	C₄	—	10	2	no	—
288	C₄	—	10	2	no	—
289	C₄	#4027	1	2	yes	—
290	C₄	C₂×D₄×D₅	1	1	yes	characteristic
291	C₄	C₂×#711	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	4	2	no	—
294	C₂	—	10	2	no	—
295	C₂	—	20	1	no	—
296	C₂	C₂×#4027	1	1	yes	characteristic
297	C₂	D₅×#131	1	1	yes	characteristic
298	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#408	C₂
prime radical	#66162	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#408	D₅
hypocenter	C₅	#2933
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#4094	C₂	semidirect
3. C₂×D₅×M₁₆	C₂	semidirect
4. C₅×#408	C₂	semidirect
5. #65408	C₂	semidirect
6. C₂×#3952	C₂	semidirect
7. #65390	C₂	semidirect
8. #65019	C₂	semidirect
9. #65018	C₂	semidirect
10. #65017	C₂	semidirect
11. #65014	C₂	semidirect
27. C₂×#681	C₂²	semidirect
29. C₂²×#168	C₂²	semidirect
33. C₅×C₂²⋊C₈	C₂²	semidirect
34. C₅×C₂²⋊C₈	C₂²	semidirect
39. #3947	C₂²	semidirect
40. #3946	C₂²	semidirect
41. #3946	C₂²	semidirect
42. C₈×Dic₅	C₂²	semidirect
77. #681	D₄	semidirect
93. #408	D₅	semidirect
159. C₂²⋊C₈	C₂×D₅	semidirect
160. C₂²⋊C₈	C₂×D₅	semidirect
176. C₂×D₅	#131	semidirect
269. C₅	#2933	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#66162	2
C₂×#4094	2
C₂×D₅×M₁₆	2
C₅×#408	2
#65408	2
C₂×#3952	2
#65390	2
#65019	2
#65018	2
#65017	2
#65014	2
#2933	5